Conversely, the signature \((\cdot ,1,A)\) , where 1 is the identity function, is easily seen to be equivalent to \((\cdot ,1,-)\) , since \(A(s)=1-s\)  for all \(s\in PT(X)\) , so modal restriction semigroups are nothing but minus-semigroups with identity. This then extends to the various other signatures. The above fairly straightforward adjustment to the definition of \(\Phi \)  in Proposition REF  does represent \(1\in A\)  correctly, and all else still works, so we could thereby recover the axiomatisations obtained in [1]}.
